Answer to Question #274848 in Calculus for sumu

Question

Your firm has recently started to give economic advice to your clients. Acting as a consultant you have estimated the demand curve of a client’s firm to be

AR = 200 - 8x where AR is average revenue ($) and x is output.

Investigation of the client firm’s cost profile shows that the marginal cost is given by: MC = x2 - 28x + 211 where MC is marginal cost ($). Further investigation has shown that the firm’s cost when no production are $10.

(a) Find equation of total cost.

(b) If total revenue is average revenue multiplied by output, find the equation of total revenue.

(c) Use the methods of differentiation, find the turning point(s) of the firm’s profit curve and say whether these point(s) are maxima or minima.

Answer 1

a)

$$TC=\int MCdx=\int (x^2 - 28x + 211)dx$$

$$=\dfrac{x^3}{3}-14x^2+211x+c_1$$

$$TC(0)=c_1=10$$

$$TC=\dfrac{x^3}{3}-14x^2+211x+10$$

b)

$$TR=(AR)x=(200-8x)x$$

$$TR=200x-8x^2$$

c)

$$P=TR-TC$$

$$P(x)=200x-8x^2-(\dfrac{x^3}{3}-14x^2+211x+10)$$

$$=-\dfrac{x^3}{3}+6x^2-11x-10, x\geq 0$$

$$P'(x)=-x^2+12x-11$$

Critical number(s)

$$P'(x)=0=>-x^2+12x-11=0$$

$$-(x-1)(x-11)=0$$

$$x_1=1, x_2=11$$

If $0\leq x<1, P'(x)<0, P(x)$ decreases.

If $1<x<11, P'(x)>0, P(x)$ increases.

If $x>11, P'(x)<0, P(x)$ decreases.

A turning point at $x=11$ is a local maximum point.

A turning point at $x=1$ is a local minimum point.